Types
of Cohomology Theory| Types
of Cohomology Theory |
De Rham Cohomology > s.a. Betti
Numbers; de Rham Theorem.
$ Def: A cohomology
theory based on p-forms 
,
and therefore only available for differentiable manifolds; Cochains are p-forms
{
p},
the duality with homology is through integration on chains, d is the exterior
derivative; Thus cocycles Zp are
closed forms, coboundaries Bp are
exact forms, and the cohomology groups are Hp(X;R):=
Zp(X) / Bp(X).
* Consequence: For an n-dimensional X, only Hp for
0 
p n can
be nontrivial.
* And homology: Hp is the dual space of Hp,
with ([],[C]):=
C .
* Ring structure: The cup product is wedge product of forms.
Cech Cohomology
@ References: Álvarez CMP(85);
Mallios & Raptis IJTP(02)
[finitary].
Equivariant Cohomology
* Applications: Kinematical understanding of topological gauge theories
of cohomological type.
@ References: Stora ht/96, ht/96.
Étale Cohomology > s.a. math
conjectures [Adams,
Weil].
* Idea: A very useful unification of arithmetic and topology.
* History: Conceived
by Grothendieck, and realized by Artin, Deligne, Grothendieck and Verdier in
1963.
@ References: Milne 79.
Floer Cohomology
@ Equivalence with quantum cohomology: Sadov CMP(95).
Sheaf Cohomology > s.a. locality
in quantum theory.
@ References: Warner 71; Griffiths & Harris 78; Strooker 78; Wells
80.
Other Types > s.a. [cohomology]; K-Theory.
@ Lichnerowicz-Poisson cohomology: de León et al JPA(97).
@ Hochschild cohomology: Zharinov TMP(05)
[of algebra of smooth functions on
torus]; Kreimer ht/05 [in
quantum field theory]; > s.a. deformation quantization.
@ Other types: Frégier LMP(04)
[related to deformations of Lie algebra
morphisms]; Papadopoulos JGP(06)
[spin cohomology].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
22 jun 2008